Question

A nutrition laboratory tested 25 "reduced sodium" hotdogs of a certain brand, finding that the mean sodium content is 310 mg with a standard deviation of 36 mg.

Construct a 95% confidence interval for the mean sodium content of this brand of hot dog and interpret a 95% level of confidence. Show all work

Answer 1

The 95% confidence interval is
`295.9 < \mu< 324.1`

A 95% level of confidence mean that there is 95% chance that the true population mean will be in this interval

Explanation:

From the question we are told that

The sample size is
`n = 25`

The mean is
`\= x = 310 \ mg`

The standard deviation is
`\sigma = 36 \ mg`

Given that the confidence level is 95% then the level of significance is mathematically represented as

`\alpha = 100 - 95`

=>
`\alpha = 5\%`

=>
`\alpha = 0.05`

Next we obtain the critical value of
`(\alpha )/(2)` from the normal distribution table , the value is

`Z_{(\alpha )/(2) } =Z_{(0.05 )/(2) } = 1.96`

Generally the margin of error is mathematically represented as

`E = Z_{(\alpha )/(2) } * (\sigma )/(√(n) )`

substituting values

`E = 1.96 * (36 )/(√(25) )`

`E = 14.1`

The 95% level of confidence interval is mathematically represented as

`\= x - E < \mu<\ \= x - E`

substituting values

`310- 14.1 < \mu< 310+ 14.1`

`295.9 < \mu< 324.1`

The 95% level of confidence mean that there is 95% chance that the true population mean will be in this interval